semi-group homomorphism and tommy's U-tetration

[Leo.W]

Such as the julia equation =

J(f(x)) = f ' (x) J(x)

then arriving at

J( f^[R](x) ) = ( f ' (x) )^R  J(x).

and 

f^[R](x) = J-inv ( f ' (x)^R  J(x) )  

which even already starts to looks like the proof for the hyperbolic.

Hi tommy
I'm not feeling like to but have to point out that, these equations aint equivalent, the Julia function denoted as J(z) fits:
\[J(f(z))=J(z)f'(z)\]
and can be arrived at
\[J(f^t(z))=J(f(f^{t-1}(z)))=J(f^{t-1}(z))f'(f^{t-1}(z))=\dot=J(z)\prod_{n=0}^{t-1}{f'(f^n(z))}\]